Accès à distance ? S'identifier sur le proxy UCLouvain
Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency
Primary tabs
- Open access
- 615.13 K
We study the nonlocal Schr"odinger-Poisson-Slater type equation −Δu+(Iα∗|u|p)|u|p−2u=|u|q−2uin ℝN, where N∈ℕ, p>1, q>1 and Iα is the Riesz potential of order α∈(0,N). We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.
Document type | Article de périodique (Journal article) |
---|---|
Access type | Accès mixte |
Publication date | 2016 |
Language | Anglais |
Journal information | "Calculus of Variations and Partial Differential Equations" - Vol. 55, no.6, p. 58 (2016) |
Peer reviewed | yes |
Publisher | Springer (Heidelberg) |
issn | 0944-2669 |
e-issn | 1432-0835 |
Publication status | Publié |
Affiliation | UCL - SST/IRMP - Institut de recherche en mathématique et physique |
Keywords | 1127 ; 1122 ; 1120 |
Links |
- Ackermann Nils, On a periodic Schr�dinger equation with nonlocal superlinear part, 10.1007/s00209-004-0663-y
- Ackermann Nils, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, 10.1016/j.jfa.2005.11.010
- Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften. 314, Springer (1996)
- Ambrosetti Antonio, On Schrödinger-Poisson Systems, 10.1007/s00032-008-0094-z
- Ambrosetti Antonio, Rabinowitz Paul H, Dual variational methods in critical point theory and applications, 10.1016/0022-1236(73)90051-7
- Bao Weizhu, Mauser N.J., Stimming H.P., Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-Xalpha Model, 10.4310/cms.2003.v1.n4.a8
- Bellazzini Jacopo, Frank Rupert L., Visciglia Nicola, Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, 10.1007/s00208-014-1046-2
- Bellazzini Jacopo, Ghimenti Marco, Ozawa Tohru, Sharp lower bounds for Coulomb energy, 10.4310/mrl.2016.v23.n3.a2
- Benci Vieri, Fortunato Donato, An eigenvalue problem for the Schrödinger-Maxwell equations, 10.12775/tmna.1998.019
- Benedek A., Panzone R., The space $L\sp{p}$ , with mixed norm, 10.1215/s0012-7094-61-02828-9
- Boas Jr. R. P., Some uniformly convex spaces, 10.1090/s0002-9904-1940-07207-6
- Bogachev Vladimir I., Measure Theory, ISBN:9783540345138, 10.1007/978-3-540-34514-5
- Bokanowski Olivier, López José L., Soler Juan, On an Exchange Interaction Model for Quantum Transport: The Schrödinger–Poisson–Slater System, 10.1142/s0218202503002969
- Bonheure Denis, Mercuri Carlo, Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials, 10.1016/j.jde.2011.04.010
- Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York (2011)
- Brezis Haim, Lieb Elliott, A Relation Between Pointwise Convergence of Functions and Convergence of Functionals, 10.2307/2044999
- Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, No. 13. Van Nostrand, Princeton, Toronto, London (1967)
- CATTO I., DOLBEAULT J., SÁNCHEZ O., SOLER J., EXISTENCE OF STEADY STATES FOR THE MAXWELL–SCHRÖDINGER–POISSON SYSTEM: EXPLORING THE APPLICABILITY OF THE CONCENTRATION–COMPACTNESS PRINCIPLE, 10.1142/s0218202513500541
- D'Aprile Teresa, Mugnai Dimitri, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, 10.1017/s030821050000353x
- Day Mahlon M., Some more uniformly convex spaces, 10.1090/s0002-9904-1941-07499-9
- Del Pino Manuel, Dolbeault Jean, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, 10.1016/s0021-7824(02)01266-7
- Di Cosmo Jonathan, Van Schaftingen Jean, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, 10.1016/j.jde.2015.02.016
- Duoandikoetxea Javier, Fractional integrals on radial functions with applications to weighted inequalities, 10.1007/s10231-011-0237-7
- Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958)
- Gilbarg David, Trudinger Neil S., Elliptic Partial Differential Equations of Second Order, ISBN:9783540411604, 10.1007/978-3-642-61798-0
- Fröhlich Jürg, Lieb Elliott H., Loss Michael, Stability of coulomb systems with magnetic fields : I. The one-electron atom, 10.1007/bf01211593
- Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14(1), 1250003, 22 (2012)
- Koskela, M.: Some generalizations of Clarkson’s inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., pp. 634–677, pp. 89–93 (1979)
- Lebedev, N.N.: Special functions and their applications, translated by Silverman RA. Prentice–Hall, Englewood Cliffs (1965)
- Le Bris, C., Lions, P.L.: From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. (N.S.). 42(3), 291–363 (2005)
- Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (118), 349–374 (1983)
- Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (2001)
- Lions P.L., Some remarks on Hartree equation, 10.1016/0362-546x(81)90016-x
- Lions P.L., The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1. * *Mp denotes the Marcinkiewicz space or weak Lp space, 10.1016/s0294-1449(16)30428-0
- Lions P. L., Solutions of Hartree-Fock equations for Coulomb systems, 10.1007/bf01205672
- Maligranda Lech, Sabourova Natalia, On Clarkson's inequality in the real case, 10.1002/mana.200610552
- Mauser N.J., The Schrödinger-Poisson-Xα equation, 10.1016/s0893-9659(01)80038-0
- Maźya, V.: Sobolev spaces with applications to elliptic partial differential equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)
- Mercuri, C.: Positive solutions of nonlinear Schrod̈inger–Poisson systems with radial potentials vanishing at infinity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(3), 211–227 (2008)
- Merle F, Peletier L.A, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The nonradial case, 10.1016/0022-1236(92)90070-y
- Milman, D.: On some criteria for the regularity of spaces of type (B). C. R. (Doklady) Acad. Sci. U.R.S.S. 20, 243–246 (1938)
- Moroz Vitaly, Van Schaftingen Jean, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, 10.1016/j.jfa.2013.04.007
- Ni Wei-Ming, , 10.1512/iumj.1982.31.31056
- Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13(3), 115–162 (1959)
- Ohtsuka Makoto, Capacité d’Ensembles de Cantor Généralisés, 10.1017/s0027763000002038
- du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, vol. 7. Oliver and Boyd, Edinburgh (1970)
- Rubin B. S., One-dimensional representation, inversion, and certain properties of the Riesz potentials of radial functions, 10.1007/bf01157392
- Ruiz David, The Schrödinger–Poisson equation under the effect of a nonlinear local term, 10.1016/j.jfa.2006.04.005
- Ruiz David, On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases, 10.1007/s00205-010-0299-5
- Slater J. C., A Simplification of the Hartree-Fock Method, 10.1103/physrev.81.385
- Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vl. 30. Princeton University Press, Princeton (1970)
- Strauss Walter A., Existence of solitary waves in higher dimensions, 10.1007/bf01626517
- SU JIABAO, WANG ZHI-QIANG, WILLEM MICHEL, NONLINEAR SCHRÖDINGER EQUATIONS WITH UNBOUNDED AND DECAYING RADIAL POTENTIALS, 10.1142/s021919970700254x
- Su Jiabao, Wang Zhi-Qiang, Willem Michel, Weighted Sobolev embedding with unbounded and decaying radial potentials, 10.1016/j.jde.2007.03.018
- Thim Johan, Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts, 10.1007/s10231-014-0465-8
- Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa. 27(3), 265–308 (1973)
- Van Schaftingen Jean, Interpolation inequalities between Sobolev and Morrey–Campanato spaces: A common gateway to concentration-compactness and Gagliardo–Nirenberg interpolation inequalities, 10.4171/pm/1947
- Willem Michel, Minimax Theorems, ISBN:9781461286738, 10.1007/978-1-4612-4146-1
- Willem Michel, Functional Analysis, ISBN:9781461470038, 10.1007/978-1-4614-7004-5
- Yang Minbo, Wei Yuanhong, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, 10.1016/j.jmaa.2013.02.062
- Yosida Kôsaku, Functional Analysis, ISBN:9783540586548, 10.1007/978-3-642-61859-8
Bibliographic reference | Mercuri, Carlo ; Moroz, Vitaly ; Van Schaftingen, Jean. Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency. In: Calculus of Variations and Partial Differential Equations, Vol. 55, no.6, p. 58 (2016) |
---|---|
Permanent URL | http://hdl.handle.net/2078.1/179187 |